This talk addresses three fundamental computational problems for algebraic ordinary differential equations: deciding the existence of solutions, eliminating differential variables, and computing rational solutions. In the first part, I will present our recent updates on the effective differential Nullstellensatz, including explicit doubly and singly exponential bounds on the number of differentiations required to decide the inconsistency of a system of algebraic differential equations. In the second part, I will focus on rational solutions of first-order algebraic differential equations from algebraic and geometric approaches.
